Phase relations and solidification behavior in the Ge-rich part of the phase diagram have been determined in two isothermal sections at 700 and 750 °C and in a liquidus projection. A reaction scheme has been derived in the form of a Schulz-Scheil diagram. Phase equilibria are characterized by three ternary compounds: τ(1)-BaRhGe(3) (BaNiSn(3)-type) and two novel phases, τ(2)-Ba(3)Rh(4)Ge(16) and τ(3)-Ba(5)Rh(15)Ge(36-x), both forming in peritectic reactions. The crystal structures of τ(2) and τ(3) have been elucidated from single-crystal X-ray intensity data and were found to crystallize in unique structure types: Ba(3)Rh(4)Ge(16) is tetragonal (I4/mmm, a = 0.65643(2) nm, c = 2.20367(8) nm, and R(F) = 0.0273), whereas atoms in Ba(5)Rh(15)Ge(36-x) (x = 0.25) arrange in a large orthorhombic unit cell (Fddd, a = 0.84570(2) nm, b = 1.4725(2) nm, c = 6.644(3) nm, and R(F) = 0.034). The body-centered-cubic superstructure of binary Ba(8)Ge(43)□(3) was observed to extend at 800 °C to Ba(8)Rh(0.6)Ge(43)□(2.4), while the clathrate type I phase, κ(I)-Ba(8)Rh(x)Ge(46-x-y)□(y), reveals a maximum solubility of x = 1.2 Rh atoms in the structure at a vacancy level of y = 2.0. The cubic lattice parameter increases with increasing Rh content. Clathrate I decomposes eutectoidally at 740 °C: κ(I) ⇔ (Ge) + κ(IX) + τ(2). A very small solubility range is observed at 750 °C for the clathrate IX, κ(IX)-Ba(6)Rh(x)Ge(25-x) (x ∼ 0.16). Density functional theory calculations have been performed to derive the enthalpies of formation and densities of states for various compositions Ba(8)Rh(x)Ge(46-x) (x = 0-6). The physical properties have been investigated for the phases κ(I), τ(1), τ(2), and τ(3), documenting a change from thermoelectric (κ(I)) to superconducting behavior (τ(2)). The electrical resistivity of κ(I)-Ba(8)Rh(1.2)Ge(42.8)□(2.0) increases almost linearly with the temperature from room temperature to 730 K, and the Seebeck coefficient is negative throughout the same temperature range. τ(1)-BaRhGe(3) has a typical metallic electrical resistivity. A superconducting transition at T(C) = 6.5 K was observed for τ(2)-Ba(3)Rh(4)Ge(16), whereas τ(3)-Ba(5)Rh(15)Ge(35.75) showed metallic-like behavior down to 4 K.

With the development of
the phonon glass–electron crystal
concept,1,2 intermetallic clathrates were considered
as promising materials for thermoelectric applications.3 Particularly clathrate type I materials have
been studied intensively, and the results have been summarized recently.4,5 Up to now, some of the most promising candidates were Ba–Ge-based
solid solutions, for which it was demonstrated that transition metals
such as Mn, Fe, Co,6 Ni, Cu,7 Zn,8 Pd, Ag,7 Cd,8 Pt, and Au7 can significantly stabilize a clathrate I phase
deriving from the binary compound Ba8Ge43□3, which exists only in a very limited temperature range from
770 to 810 °C.9 Hitherto, the existence
of a clathrate I Ba8RhxGe46–x–y□y was not reported. Thus, the aim of this
investigation is (i) to explore the formation and homogeneity range
of the corresponding clathrate I phase, (ii) to establish the phase
relations in the Ge-rich part of the Ba–Rh–Ge system,
(iii) to characterize the crystal structure of new ternary phases,
and (iv) to provide information on the physical properties of the
compounds involved. The results of density functional theory (DFT)
calculations are included in order to analyze the trend of thermodynamic
stability and the change of the density of states (DOS) as a function
of Rh doping in the clathrate I phase Ba8RhxGe46–x.

2
Experimental Details

2.1
Sample Preparation and Characterization

Samples for phase analyses and crystal structures were prepared
by argon arc melting from elemental ingots (Ba, 99.9 mass %; Rh, 99.99
mass %; Ge, 99.999 mass %) on a water-cooled copper hearth. To compensate
for losses of Ba, ∼2 wt % was added beforehand. For all samples,
the weight loss was below 2 wt %. The arc-melted buttons were cut
into two pieces and sealed in evacuated quartz tubes. All samples
were annealed for 7 days at 750 and 700 °C and afterward characterized
by powder X-ray diffraction (PXRD), differential thermal analysis
(DTA), and electron probe microanalysis (EPMA). For PXRD, a Huber
imaging-plate system equipped with a monochromator (Cu Kα1 radiation) was used. A detailed evaluation of the powder
patterns was performed via Rietveld refinements employing the FullProf program.10 Scanning
electron microscopy images and EPMA measurements using an energy-dispersive
X-ray (EDX) detector (Oxford Instruments) were carried out on a Zeiss
Supra 55VP instrument. Single-crystal X-ray diffraction data at 300
K (for the clathrate I phase also at 200 and 100 K) were collected
on a four-circle Nonius Kappa diffractometer equipped with a CCD area
detector and monochromated Mo Kα radiation, whereby isothermal
temperatures for the crystal, mounted with transparent varnish on
a glass rod, were assured by a continuous stream of nitrogen gas enclosing
the crystal at a preset temperature. Single-crystal X-ray data were
analyzed using the SHELX program system.11 DTA measurements were performed on Netzsch DSC
404C equipment under an argon atmosphere with an uncertainty in the
temperature of ±5 °C.

The single-phase samples for
physical property measurements (∼2 g each) were synthesized
in the same way as those described above (BaRhGe3, as cast)
and annealed at 800 °C (clathrate I) and 770 °C (Ba3Rh4Ge16 and Ba5Rh15Ge35.75), respectively. Resistivity measurements at various
magnetic fields were performed in a Cryogenic 3He bath
top-loading cryostat with a maximum field of about 12.5 T. The accessible
temperature covers the range from 300 mK to room temperature. The
samples were mounted on a platform, with the current flowing parallel
to the applied magnetic field using a four-probe technique. The resistivity
was measured using a high-precision Lakeshore bridge with a 13.7 Hz
alternating-current technique and 10 mA excitation current. Specific
heat measurements were performed on a PPMS Quantum Design 6000 with
the relaxation calorimetry technique from 2 to 20 K in magnetic fields
of 0–2 T. An additional zero-field specific heat measurement
was performed in the temperature range from 3 to 80 K employing an
adiabatic step-heating technique (∼1 g sample mass). At temperatures
above 300 K, the electrical resistivity and Seebeck coefficient were
measured simultaneously with an ULVAC ZEM3 system.

2.2
DFT Calculations

DFT calculations
were carried out by the Vienna ab initio Simulation Package.12 The exchange-correlation functional was parametrized
in terms of the local density approximation according to Ceperley
and Alder.13 The electron–ion interaction
is treated within the framework of the projector augmented wave method.14,15 The valence state configuration for construction of the pseudopotentials
included the 5s, 5p, and 6s states for Ba, the 5s and 4d states for
Rh, and the 3d, 4s, and 4p states for Ge. For structural relaxations,
i.e., total energy minimization and relaxation of atomic forces, a
5 × 5 × 5 k-point grid according to Monkhorst
and Pack16 resulted in well-converged structural
parameters. For calculations of the electronic DOSs, a dense 11 ×
11 × 11 k-point mesh was constructed.

3
Binary Boundary Systems

For the Ge-rich
part (>70 atom % Ge) of the Ba–Ge system,
the phase diagram reported by Okamoto17 was accepted, which was adapted from refs (18) and (19). The Rh–Ge boundary
system (≥50 atom % Ge) was taken from Massalski.20 The crystallographic data of all relevant phases
are listed in Table 1. No contradictions to
the accepted binary boundary systems were observed throughout all
of the investigations carried out within this work.

4
Results and Discussion

4.1
Clathrate I Solid Solution at 800 °C

The formation and solubility limits of the clathrate I phase Ba8RhxGe46–x–y□y at 800 °C were studied by PXRD and EPMA on samples with nominal
compositions x = 1–4 and y = 0. In all cases, a successful indexation of the clathrate I phase
with the crystal structure of K4Ge23–x was possible. The sample with x = 1 revealed Ge as the only secondary phase; all other samples (x ≥ 2) revealed the presence of Ge and an increasing
amount of τ2with increasing Rh content. Careful
analysis of the lattice parameters and the EPMA data revealed a solubility
range of Rh in Ba8Ge43□3 extending
into the ternary region up to the composition Ba8Rh1.2Ge42.8□2.0 (see Figure 1). In comparison to the isovalent Co, for which
a maximum solubility of 2.5 atoms/unit cell was reported,6 the solubility of the larger Rh atoms (rRh = 0.1345 nm and rCo = 0.1252 nm21) decreases significantly
but leads to a typical enlargement of the cubic lattice. The Ba content
of the sample Ba8Rh1Ge45 is, according
to EPMA data, slightly higher than that for the other samples, indicating
a higher concentration of vacancies (□). Consistent with this
observation, Rietveld refinements of the PXRD patterns—fixing
the Rh content to the EPMA data—result in a composition of
Ba8Rh0.6Ge43□2.4 for the sample with the smallest Rh content and Ba8Rh1.2Ge42.8□2.0 for all others exceeding
the solubility limit. EPMA data of the as-cast alloy Ba8Rh1Ge45 result in a Rh content of 1.7 atom
%, whereas during annealing at 800 °C, the Rh content shrinks
to 1.2 atom % and superstructure reflections of Ba8Ge43□39 appear clearly
observable in the powder pattern. With respect to clathrate type I
(Pm3̅n, a0), Ba8Ge43□3 crystallizes in an 8 times
larger unit cell (a = 2a0) with space group Ia3̅d because
of a full order among Ge atoms and vacancies. For details, see ref (9). Rietveld refinements of
the superstructure cell revealed a Rh content of 0.7 atoms/formula
unit in the 24c position, which remains completely vacant in the fully
ordered structure of binary Ba8Ge43□3. This result is in sound agreement with the 0.6 Rh atoms/formula
unit found by EPMA. The as-cast sample with EPMA composition Ba8Rh0.9Ge43.1□2 does
not show superstructure reflections, indicating the breakdown of the
vacancy ordering and a random mixture of all three species (Rh, Ge, and □)
and thus a reduction of the crystal symmetry to the conventional clathrate
I structure with the smaller unit cell (a0). This observation may indicate that the substitution of Ge atoms
by Rh atoms, instead of exclusively filling vacant positions, starts
between a Rh content of 0.6 and 0.9 atoms/formula unit (highlighted
by the blue vertical line in Figure 1). The
phase transition might be of second order because the symmetries of
the structure and superstructure exhibit a crystallographic group–subgroup
relation. Interestingly, all samples reveal a slightly higher Rh content
in the clathrate phase in as-cast conditions (see Figure 1), indicating that the temperature of 800 °C
is not the temperature at which the maximum solubility of Rh is reached
in the clathrate I phase.

4.2
Phase Equilibria in the Ternary System Ba–Rh–Ge
(>60 atom % Ge)

4.2.1
Phase Equilibria at 700 °C

The phase relations at 700 °C based on PXRD as well as EPMA
data are summarized in Table S1 (Supporting Information). Corresponding micrographs representing the three-phase equilibria
are shown in Figure 2. The isothermal section
is drawn in Figure 3. Only one ternary phase,
τ1-BaRhGe3 crystallizing in the BaNiSn3 structure type,22 exhibits a solubility
range (defined by EPMA measurements). Because the Ba content in τ1-BaRhGe3 for all measurements does not deviate
significantly from 20 atom %, partial substitution of Rh by Ge in
the 2a site rather than vacancy formation is suggested (BaRh1–xGe3+x), although in this
case, the Rietveld refinement of the multiphase samples does not allow
any conclusion on substitution or vacancy formation. The maximum solubility
of Ge was detected in an alloy located in the three-phase region with
τ2 and κIX, being x = 0.22. The lattice parameter a is slightly increasing
with increasing Ge content, whereas c is slightly
decreasing. Compared to the lattice parameters of the as-cast τ1-BaRhGe3 (stoichiometric composition), the unit
cell volume is increased by 2%. The existence of the fully ordered
structure, reported earlier,22 was not
investigated here. For the binary phases Rh17Ge22 and κIX, the solubility of the third element was
below the detection limits. The three-phase equilibrium (Ge) + κIX + τ2, as observed in sample 6 (see Figure 2), clearly documents the fact that a type I clathrate
phase does not form at 700 °C. The low-temperature phase, BaGe5, reported by Aydemir et al.,23 was not observed in this investigation.

4.2.2
Phase Equilibria at 750 °C and Invariant
Reactions up to 830 °C

In contrast to the isothermal
section at 700 °C, the clathrate I phase Ba8RhxGe46–x–y□y exists at 750
°C as a truly ternary phase and extends to a maximum Rh content
of xmax = 1.2 in equilibrium with τ2 and κIX (see Figure 4). The solubility limit is practically the same as that at 800 °C.
Because the existence of binary Ba8Ge43□3 is limited to a temperature interval from 770 to 810 °C,9 also a lower limit of the solubility of Rh is
expected and indicated in the isothermal section (Figure 4) but was not investigated. Rh stabilizes the clathrate
I structure to 740 °C, where it decomposes in terms of a eutectoid
reaction, κI ⇔ (Ge) + κIX +
τ2, as determined by DTA measurements. Furthermore,
EPMA data indicate a small solubility of Rh (0.5 atom %) in the clathrate
IX phase. Similar to the results at 700 °C, a solubility range
for BaRh1-xGe3+x with xmax = 0.25 was found. The crystallographic data
and compositions of the compounds for the three-phase fields are presented
in Table S2 (Supporting Information), and
the respective micrographs are shown in Figure 5.

Careful analyses of various samples in the as-cast
state as well
as after annealing at 820, 800, 775, 770, 750, and 700 °C combined
with DTA measurements revealed a rather complex reaction scheme between
830 and 700 °C, yielding nine four-phase reaction isotherms (see
Figure 6). The corresponding liquidus and solidus
projections are illustrated in Figure 7. In
Figure 8, selected micrographs of different
compositions are presented. Figure 8a documents
a backscatter image of sample 7 annealed at 800 °C. The DTA measurement
of this alloy shows the liquidus to appear slightly above 800 °C.
From the micrograph, huge primary crystals (diameter >40 μm)
of Rh17Ge22 can be seen. Those are surrounded
by the peritectically formed τ3 phase (P1 in Figure 6), and finally the quenched liquid
exhibits a fine eutectic structure containing three phases [(Ge) + κI + τ3; see also Figures 7 and 6, E1]. From the as-cast
sample 12, a cascade of several reactions can be examined (Figure 8b). The joint crystallization of τ1 and τ3 is followed by the peritectic formation
of τ2 (P2 in Figure 6), and finally the crystallization field of the clathrate I phase
is entered. In contrast to sample 4, the micrograph of sample 3 (Figure 8) shows that the (Ge)
grains have much sharper boundaries because the sample position is
still in the primary crystallization field of (Ge). Although clathrate
I exhibits a very narrow crystallization field of only 2 °C,17,18 this field is opening in the ternary system to a wide temperature
and compositional range (Figure 7). A representative
micrograph of sample 4 clearly shows the large primary grains of κI. When the solidus (Figure 7), the
isothermal section at 750 °C (Figure 4), and DTA are compared, two solid-state transition-type reactions
are suggested around 760 °C (U4 and U5 in Figure 6). Because
diffusion in solid-state reactions usually is dramatically reduced
compared to reactions in which a liquid phase is involved, the EPMA
data as well as PXRD patterns of samples 1 and 2 at 750 °C reveal
the presence of small amounts of nonequilibrium phases τ3 and κI, respectively.

4.3
Crystal Structures of the Ternary Phases

4.3.1
Crystal Structure of the Clathrate I Phase

A suitable single crystal of the clathrate I phase was selected
from a sample with the maximum Rh content annealed at 800 °C.
Analysis of the measured X-ray intensity data revealed isotypism with
the cubic crystal structure of K4Ge23–x (clathrate I type, space group Pm3̅n). The occurrence of the necessary splitting
of the 24k position (Ge2) in Ge-based type I clathrates indicates
the presence of vacancies in the neighboring 6d position, as observed
also in other clathrates.24,25 As in the case of Ba8Rh1.2Ge42.8□2.0, three
different species (Ge, Rh, and □) are occupying the 6d site,
the Rh content was fixed during the structure refinement to an
EPMA value of 1.2 Rh atoms/unit cell. The atomic displacement parameters
(ADPs) of the other framework positions (16i and 24k) did not infer
a mixed occupancy of Rh and Ge. Thus, all Rh atoms were placed in
the 6d position, and the Ge content was allowed to vary during the
refinement. The results presented in Table 2 [interatomic distances are listed in Table S3 (Supporting Information)] demonstrate sound correspondence
of the vacancy amount in the 6d position with the occupancy of the
Ge22 atoms in the 24k position at closer distances to the vacancies.

All attempts to allow anisotropic refinement of the
ADPs at the
24k split position for the data set collected at 200 K led to unstable
results, and therefore these positions were refined considering isotropic
displacement parameters only. As summarized for several other clathrate
I compounds, analysis of temperature-dependent ADPs of guest and framework
atoms allows conclusions on the lattice dynamics of the respective
compound.4Applying least-squares fits
to the equations given by Christensen4 to
extract Debye (θD) and Einstein (θE) temperatures resulted in θD = 245 K, θE,11 = 88 K, and θE,22,33 = 70 K. These values
are well within the range of the values for Ba–Ge-based clathrate
I compounds reported earlier.4,26 The two Einstein temperatures
originate from symmetry constraints (U11 is different from U22 = U33 for the 6c site). The Debye temperature was calculated
by taking into account only ADPs of the 16i position because this
crystallographic site is the only framework position unaffected by
mixed occupancy or splitting. Applying the calculation method tested
earlier26 to evaluate the thermal expansion
coefficient α from the temperature-dependent lattice parameters a(T), α100–300K= 14.0 × 10–6 K–1 was found.
The results of these analyses are displayed in Figure 9.

4.3.2
Crystal Structure of the Phase τ2-Ba3Rh4Ge16

A suitable
single crystal was selected from a sample with a nominal composition
of τ2 annealed at 775 °C for 1 week. Indexation
of the single-crystal X-ray diffraction data revealed a tetragonal
crystal lattice with a = 0.65643(2) nm and c = 2.20367(8) nm. Analysis of the systematic extinctions
prompted the extinction symbol I--- with the centrosymmetric
body-centered-cubic space group I4/mmm with highest symmetry. The structure was solved by applying direct
methods and refinement of seven atom sites with anisotropic ADPs converged
to RF = 0.0273 and a residual electron
density of <2 e/Å3. Detailed results are listed
in Table 3 and atom parameters are given in Table 4. [Interatomic distances are listed in Table S4 (Supporting Information)]. Ba atoms are located at the 2a
and 4d positions, Rh occupies the 8g site, and Ge atoms adopt the
16m, 8j, and two 4e sites. During the refinement, full ordering
and full occupancy of all sites was confirmed. The composition extracted
from the single-crystal analysis is in fair agreement with the EPMA
results (see Table 3).

The unit cell dimensions exhibit a close relation
to the BaAl4 structure type, crystallizing in the same
space group symmetry: a = ∼√2aBaAl4 and c = ∼2cBaAl4. The 4-fold cell volume is
caused by occupation of
the voids and some significant shifts of the atomic sites with respect
to the BaAl4 structure type. The positions of the Ba atoms
are split but principally maintained; the Al1 site splits into the
Ge1 and Ge2 sites. Half of the Al2 sites are occupied by the Ge3 and
Ge4 atoms, while the other half is shifted along [001] and occupied
by Rh atoms, which allows a new site for the Ge2 atoms and significantly
different coordination figures around the two Ba sites. The coordination
polyhedron of the Ba2 atoms in the 4d site with coordination number
18 is akin to the coordination polyhedron of Ba in the BaAl4 structure; only a slight distortion is observed as the Al atoms
at the 4e site of BaAl4 are replaced by two 4e sites, both
occupied by Ge atoms. In addition, the Rh site is located on the top
and bottom of the polyhedron, forming tetragons with the Ge1 atoms
(see Figure 10). Analysis of the Voronoi coordination polyhedron of Ba1 using the
program DIDO95(27) defined
24 neighboring atoms with areas contributing to the Dirichlet domain
(Ba1[Ge28Ge18Rh8]; Figure 10). Infinite and alternating layers of polyhedra
around the Ba1 and Ba2 atoms describe the whole crystal structure:
the polyhedra around the Ba2 atoms are connected via hexagons to other
four neighboring and symmetry-equivalent polyhedra, and also the polyhedra
around the Ba1 atoms are within a layer connected via tetragonal faces
to four neighbors. The crystal structure is built up by an alternation
of those different layers, which are sharing trigonal faces and leaving
tetragonal-pyramidal cavities (see Figure 10). The search for an identical structure type (Pearson symbol: tI46)
in Pearson’s Crystal Data28 prompted
no results, defining Ba3Rh4Ge16 as
a unique structure type.

4.3.3
Crystal Structure of the Phase τ3-Ba5Rh15Ge36-x

A
suitable single-crystal fragment of the phase was selected from an
alloy [sample 15; Ba, 7.5 atom %; Rh, 30 atom %; Ge, 62.5 atom%; see
also Table S1 (Supporting Information)]
annealed at 800 °C. X-ray diffraction data were successfully
indexed on the basis of a large orthorhombic unit cell exhibiting
systematic extinctions consistent with face-centering and lattice
dimensions of a = 0.84570(2) nm, b = 1.4725(2) nm, and c = 6.644(3) nm. Symmetry analysis
revealed space group Fddd as the only possible space
group from the extinction rules. Direct methods allowed the assignment
of the three different species (Ba, Rh, and Ge) to the respective
crystallographic positions, resulting in reasonable interatomic distances
[see Tables 3, 5, and S5 (Supporting
Information)] and sound agreement of the refined composition
Ba9.0Rh26.9Ge64.1 with the EPMA value
of Ba8.9Rh27.2Ge63.9.

Whereas
refinement of the occupancy for Ge2 resulted in a slight deficiency
(occ = 0.93), the same attempt failed for Ge1 located in the 32h position
with x = 0.2528, y = 0.2531, and z = 0.2929. When isotropic ADPs were applied, two rather
high peaks of ∼15 e/Å3 were detected by difference
Fourier synthesis at a distance closer than 0.05 nm to the respective
atom position. A detailed inspection of the electron density at this
site via a difference Fourier calculation by removing the Ge1 atom
revealed an elliptically shaped electron density in the two-dimensional
plot parallel to the (001) plane (Figure 11). Although the three-dimensional image does not provide a clear
conclusion on the splitting of this site, the improvement in the R values after splitting into two or three separate positions
is significantly large (from 0.07 to 0.034 for RF in both cases). The splitting into a 2-fold site with isotropic
ADPs leaves a small electron density at a distance of 0.058 nm to
the Ge position with higher occupation (69%) but with no correlation
to the vacancy amount of Ge2. In contrast to this observation, a 3-fold
splitting results in reasonable Uiso values,
and even without restraining the sum of the occupancies to unity,
it results in a completely filled 32-fold site, indicating that the
whole electron density in this particular part of the unit cell is
covered, allowing three atom positions. Moreover, the distribution
of the occupancy [Table 5 and Figure S1 (Supporting Information)] over the three sites
correlates with the vacancy amount at the Ge2 site. The sites Ge1a/b/c
are located inside an infinite channel formed by Ge pentagons and
Rh triangles [see Figure S1 (Supporting Information)]. Ge1a is located directly in the center of the triangle formed
by Rh atoms, and all bonding distances are rather short compared to
the other Ge–Rh distances in this structure [see Table S5 (Supporting Information)]. It seems that, whenever
one of the Ge2 sites remains vacant (occ. 7%), the two Ge1 atoms close
to this vacancy are shifted further away to the Ge1b site (occ. 15%),
closer to the pentagonal plane formed by two Ge6, two Ge7, and one
Ge9, respectively [Figure S1 (Supporting Information)]. In order to avoid the short distances to the Rh atoms, the third
split site (Ge1c position) is released to the other side of the triangle
closer to the pentagonal face formed by two Ge2, two Ge4, and one
Ge10 [see Figure S1 (Supporting Information)].

The final refinement with anisotropic ADPs (except
for Ge1a/b/c)
converged to RF = 0.0340 and a residual
electron density of <2.3 e/Å3. Detailed results
are listed in Table 3. There are three distinct
positions for Ba atoms present in the crystal structure, and all of
their coordination polyhedra are characterized by distorted faces
(see Figure 12). Although the Voronoi polyhedra
of Ba1 and Ba2 look similar, a detailed analysis of the Dirichlet
domains using the program DIDO95(27) revealed a slight difference in the coordination number:
whereas Ba1 has 20 neighbors [neglecting two Ba2 atoms, which are
contributing an area of only 1% to the domain (see the Supporting Information)], for Ba2, only 19 neighbors
were identified [also neglecting two Ba1 atoms, which are contributing
a small area (1.1%) to the domain; see Table S5 (Supporting Information) and Figure 12]. The entire structure is built up by blocks, which are formed by
layers of polyhedra around the Ba1 and Ba2 atoms. They are alternately
linked to each other by sharing pentagons. Between two of those layers the polyhedra around the Ba3 atoms are located, which are not connected
via any face or atom but are linked to four neighbors via bonds to
two Rh4 and two Rh5 atoms. The Ba3 polyhedra are linked to four Ba1
and four Ba2 polyhedra, two of each at the bottom and top sharing
tetragons. The layers of Ba1/Ba2 polyhedra are rotated against each
other by 60°. The four blocks per unit cell are shifted and rotated
according to the symmetry operators of the space group Fddd. A search for the structure type (Pearson symbol: oF448) in Pearson’s
Crystal Data28 prompted no results with
these lattice parameters, crystal symmetry, and Wyckoff sequence,
suggesting Ba5Rh15Ge35.75 to be a
unique structure type.

4.4
DFT Results

The energies of formation
of Ba8RhxGe46–xfor various values of x = 0–6
were derived from the DFT calculations at zero pressure, which are
shown in Figure 13. As mentioned in our previous
studies, experimentally the dopants are randomly distributed over
the 6d site. For the DFT calculations, the doping atoms have to be
placed on specific sites within the unit cell, which, in general,
causes changes of the lattice symmetry and cell shape. Therefore,
to be consistent with the experiment, a cubic unit cell was enforced
for all calculations. More details on the types of calculations are
given in refs (29) and (30). For comparison, the enthalpies
of formation (ΔH) for Ba8PdxGe46–x and Ba8AgxGe46–x,29 which were studied
recently, are also shown in Figure 13. The
incorporation of Ag into the Ge framework stabilizes the clathrate
I compound, as evidenced by a decrease of ΔH of −10 kJ/mol for x = 0–5 (Figure 13). Further enrichment of Ag up to Ba8Ag6Ge40 does not further decrease ΔH, which is in line with the solubility limits found in
the experiments. In contrast to this, alloying of Rh and Pd has much
more pronounced effects on the stabilization of Ba8MxGe46–x. The formation energies for both compounds decrease monotonically.
These unequal behaviors are attributed to the difference in pd bonding
between closed and open d shell elements, which were discussed in
our previous work.31

DOSs for Ba8RhxGe46–x (with x = 1–6) are shown
in Figure 14. Significant changes by doping
with Rh can be observed. The gap,clearly present in Ba8Ge46, shrinks more and more by substituting Ge by Rh,
implying strong hybridization between the two framework-forming species.
For the hypothetical compound Ba8Rh1Ge45, a peak emerges in the gap. As the doping of Rh increases, the peak
extends and gradually mixes with framework states, which leads to
small pseudogaps at about −0.1 eV in Ba8Rh3Ge43 and at about 0.3 eV in Ba8Rh4Ge42, respectively. Upon further substitution of up to
six Rh/unit cell, a gap appears. In this case, it is located above
the Fermi level.

Our previous studies29−31 revealed that the location
of EFplays an important role in understanding
the
Seebeck coefficient. For Ba8MxGe(Si)46–x, a simple electron-counting
rule was proposed to define the critical composition, for which the
Fermi energy falls into the gap, i.e., xgap = 16/(4 – n), with n being
the valency of the dopant. This rule has been successfully applied
to Ba8MxGe(Si)46–x (M = Cu, Ag, Au, Pd, and Ni). Assuming a formal
valency of −1 for Rh results in xgap = 3.2. This value of −1 derives from the electron configuration
of the outer shell (Rh, 4d85s; Ge, 3d104s24p2). In order to arrive at the isovalent composition,
which corresponds to the Sn configuration (4d105s25p2),Rh requires five additional electrons. Because the
valency, n, of Ge is 4, the five less electrons for
Rh results in n = −1. Please note that each
vacancy (□) corresponds to a valency n = 0
because no electrons are provided. In Figure 14, a narrow valley in the DOS of Ba8Rh3Ge43 at about −0.1 eV is observed, but a real gap is absent.
Experimental samples are Ba8RhxGe46–x–y□y with a solubility of Rh up
to about 1.2. This indicates the importance of vacancies in the stability
and bonding of the clathrate I structure. If one assumes that each
vacancy compensates four electrons, xgap = 1.6 is obtained for a semiconducting charge-balanced compound
Ba8RhxGe46–x–y□y, which is much closer to the experimental value for the solubility
of Rh in the present system.

4.5
Physical Properties

4.5.1
Physical Properties of the Clathrate I Ba8Rh1.2Ge42.8□2.0

The temperature dependence of the electrical resistivity (ρ)
as well as of the Seebeck coefficient (S) of a clathrate
sample with a maximum Rh content of 1.2 atoms/unit cell is plotted
in Figure 15. PXRD and EPMA data for this sample
prompted minor impurities of Ge (<4 wt %) and τ2 (<2 wt %). Both electrical resistivity and the Seebeck coefficient
reveal an almost linear temperature dependence typical for metallic-like
systems, which is expected for Ba8Rh1.2Ge42.8□2.0 in terms of the Zintl formalism.
The charge-balanced composition is Ba8Rh3.2Ge42.8 (Rh compensates for 5e– in the tetrahedrally bonded Ge
framework), but because the Rh solubility limit at 800 °C is
1.2 and each vacancy is considered to compensate for 4e–, there remain
still 2e– of the total 16e– supplied by the Ba atoms (eight Ba2+).

In order to get a rough estimation of the charge
carrier density,
Mott’s formula 1 was employed, giving
the diffusion thermopower in terms of a free electron model32

[Formula ID: eq1]1

where me is the
electron mass and e is the carrier charge. A linear
fit to S(T) resulted in a charge
carrier density n of ∼1021 cm–3. This value is in rather sound agreement with 1.6
× 1021 cm–3, resulting from a simple
calculation assuming that the two uncompensated electrons/unit cell
[a = 10.6932(2) nm] for Ba8Rh1.2Ge42.8□2.0 serve as charge carriers.

4.5.2
Physical Properties of τ1-BaRhGe3

An as-cast sample was used for electrical
resistivity measurements. EPMA analysis consistent with the Rietveld
refinement proved BaRhGe3 as the main phase with traces
of BaRh2Ge2 (<3%; ThCr2Si2 type33). Below 300 K, BaRhGe3 exhibits a temperature dependence of the electrical resistivity
typically observed for metallic compounds (see Figure 16). Down to 1.8 K, no superconducting transition is observed
and ρ(T) can be described up to 300 K fairly
well using the Bloch–Grüneisen model (ρBG in eq 2) for metals (red solid line in Figure 16), resulting in a residual resistivity ρ0 = 54 μΩ cm and a Debye temperature θD = 193 K.

4.5.3
Physical Properties of τ2-Ba3Rh4Ge16

The electrical
resistivity of Ba3Rh4Ge16 was measured
in the temperature range from 300 mK to 300 K. The resistivity vanishes
at TC = 6.5 K, indicating a transition
into a superconducting state. Above TC, the resistivity data exhibit a simple metallic temperature dependence.
The residual resistivity ratio from room temperature to 6.5 K is around
6.2, referring to a good sample quality. The s–d scattering
on Rh atoms in the sense of Mott and Jones has been taken into account
but does not seem to possess any significant contribution. However,
the results can be accounted for in terms of the parallel resistor
model:34

[Formula ID: eq2]2

In this model, ρBG is the Bloch–Grüneisen term and ρmax the saturation resistivity. A least-squares fit reveals a residual
resistivity of ∼23 μΩ cm and a saturation resistivity
of ∼321 μΩ cm (shown as a solid line in Figure 17). As a result, a Debye temperature of ∼145
K was determined. Employing the empirical model of McMillan leads
to an electron–phonon coupling constant λep ≈ 0.9, indicating a value beyond the BCS weak-coupling limit.
Resistivity measurements were carried out in various external fields
also. With increasing magnetic field, the transition temperature shifts
to lower values until the superconducting state is completely suppressed
at 1.5 T in the temperature range available (inset Figure 17). A commonly used 90/10 technique, i.e., the midpoint
between 90% of ρ0 and 10% of ρ0,
determines the superconducting transition temperature. Results are
collected in Figure 18. The initial slope of
the upper critical field is derived as

[Formula ID: eq3]3

The model by Werthamer et al.35 (WHH) is used to describe the temperature dependence of
the upper critical field (red solid line in Figure 18). This model takes into account the spin–orbit scattering
(λso), causing the electron spin to no longer be
a good quantum number, and the depairing mechanism due to Zeeman splitting,
called the Pauli limiting (PL). Quantitatively, two parameters guide
this model, i.e., the Maki parameter36 and
the spin–orbit interaction constant. The Maki parameter can
be estimated from

[Formula ID: eq4]4

This small value hints to a rather limited
influence of the PL and, therefore, the establishment of orbital depairing
to play the essential role in this material. The WHH model predicts
an upper critical field of 1.14 T. An alternative route to derive
α follows from35

[Formula ID: eq5]5

where e is the electron charge, me the electron mass, γ the Sommerfeld value, and
ρ0 the residual resistivity. The Maki parameter obtained
from eq 5 coincides very well with that derived
from eq 4. Details on the superconducting state
of τ2-Ba3Rh4Ge16 will be presented in a future publication.

Heat Capacity of τ2-Ba3Rh4Ge16

Specific heat measurements
were performed from 2 to 80 K in magnetic fields from 0 to 2 T, and
the results are displayed as Cp/T versus T in Figure 19. In zero fields, a sharp second-order transition occurs at
around 6.7 K, confirming the superconducting transition. In conjunction
with the heat capacity anomaly at T = TC, bulk superconductivity can be concluded. The electronic
Sommerfeld coefficient (γ) is about 80 mJ/mol·K2. The Hc2 behavior extracted from the
field dependence of the superconducting specific heat anomaly fits
well to the results obtained from electrical resistivity measurements.
The ratio of the superconducting specific heat jump to the normal
state heat capacity γTC yields

[Formula ID: eq6]6

which is close to the BCS value of 1.43. The
inset of Figure 19 shows the field dependence
of the jump at TC.

4.5.4
Physical Properties of τ3-Ba5Rh15Ge35.75

The electrical
resistivity from 4 to 300 K is plotted in Figure 20. The temperature dependence is metallic-like, and in contrast
to Ba3Rh4Ge16, no superconductor
transition was observed to 4.1 K. At elevated temperatures, the slope
of the electrical resistivity significantly deviates from the simple
linear temperature dependence and can be well described by applying
the parallel resistor model (eq 2). Using this
model to fit the data, residual resistivity ρ0 =
320 μΩ cm, saturation resistivity ρmax= 784 μΩ cm, and Debye temperature θD = 150 K were extracted. Measuring the Seebeck coefficient at room
temperature revealed electrons to be the main charge carriers and
that the value of −6 μV/K is relatively small, as expected
for metals and intermetallic compounds.

5
Conclusions

Although the solubility
of Rh is rather limited, Rh was found to
stabilize a clathrate phase of type I, Ba8RhxGe46–x–y□y. The homogeneity
range at 800 °C extends to a Rh content of 1.2 atoms/unit cell
at a remaining vacancy content of y = 2, accompanied
by an increase of the lattice parameter to a = 1.06932(2)
nm. Investigating the lattice dynamics of this compound using single-crystal
X-ray data revealed features similar to that already reported for
other Ge-based clathrates.4 Phase relations
in the Ge-rich part of the Ba–Rh–Ge system are characterized
by the existence of two novel compounds, Ba3Rh4Ge16 and Ba5Rh15Ge35.75, for both of which a new, unique crystal structure was established
by single-crystal X-ray measurements.

The DFT-derived enthalpies
of formation for Ba8RhxGe46–x support the experimental results,
namely, that the incorporation
of Rh atoms into the clathrate I framework significantly stabilizes
the compound. The DOSs reveal gaps or deep minima, which are linked
to the large gap of Ba8Ge46 about 1 eV below EF. The Fermi energy gets closest to a deep minimum
for x = 3, in line with the valence electron-counting
rule, which yields xgap = 16/5 for the critical concentration.

The limited homogeneity
range of Ba8RhxGe46–x–y□y prevents this compound
from exhibiting interesting thermoelectric properties because the
sample with the maximum Rh content, Ba8Rh1.2Ge42.8□2.0, still has metallic properties
and thus the Seebeck coefficient of −20 μV/K at room
temperature remains below thermoelectrically interesting values. The
electrical resistivities of BaRhGe3 and Ba5Rh15Ge35.75 increase with increasing temperature,
exhibiting metallic behavior, and for Ba5Rh15Ge35.75, revealing a 6 times higher residual resistivity
ρ0 (320 μΩ cm), a tendency for saturation
above 300 K is observed. Ba3Rh4Ge16 shows a superconducting transition at a critical temperature of TC= 6.5 K and the upper critical field is completely
suppressed at 1.5 T, close to the predicted value of 1.14 T from the
WHH model. The heat capacity data confirmed bulk superconductivity
and BCS-like behavior.

Supporting Information Available

X-ray
crystallographic data
in CIF format and additional figures and tables. This material is
available free of charge via the Internet at http://pubs.acs.org.

Supplementary Material

Notes

The authors declare no
competing financial interest.

Acknowledgments

The authors acknowledge the contribution of H. Fasshuber,
solving the crystal structure of Ba5Rh15Ge35.75 within a student’s research project at the Institute
of Physical Chemistry, University of Vienna. M.C. and R.P. gratefully
acknowledge support by the Austrian Science Foundation FWF under Project
P22295-N20.

Compositional dependence of the lattice parameters
for Ba8RhxGe46–x–y□y quenched from 800 °C, compared to as-cast data. The lattice
parameters of Ba8Ge43□3 and
for the compound with x = 0.6 were taken as a/2 of the superstructure of clathrate I.9 The blue vertical bar indicates the symmetry change from
the superstructure to the clathrate I structure.

[Figure ID: fig2]

Figure 2

Micrographs (backscatter
detector) of selected samples at 700 °C.
The corresponding data are given in Table S1 (Supporting Information); sample positions are indicated by
black circles in Figure 3.

Micrographs (backscatter detector) of selected samples
at 750 °C.
The corresponding data are given in Table S2 (Supporting Information), and black circles in Figure 4 indicate the sample positions.

[Figure ID: fig6]

Figure 6

Reaction scheme (Schulz–Scheil
diagram) for the Ge-rich
part of the Ba–Rh–Ge system in the temperature range
from 830 to 700 °C.

[Figure ID: fig7]

Figure 7

Liquidus and solidus projections of the Ge-rich part of
the Ba–Rh–Ge
system. The liquidus projection and primary crystallization fields
of the phases are colored red, and the solidus, phases, and sample
positions and numbers are colored black. The blue labels indicate
the invariant reactions listed in Figure 6.

[Figure ID: fig8]

Figure 8

Selected micrographs (backscatter detector) of several
samples
(sample 7 was annealed at 800 °C for 1 week; all others are in
the as-cast state after arc melting).

[Figure ID: fig9]

Figure 9

ADPs Uij and lattice
parameter a as a function of the temperature for
Ba8Rh1.2Ge42.8□2.0.

[Figure ID: fig10]

Figure 10

Crystal structure of τ2-Ba3Rh4Ge16. Ba atoms are colored blue,
Rh green, and Ge yellow.
In the projection on the (001) plane, the respective unit cell of
the BaAl4 type is outlined by blue lines.

[Figure ID: fig11]

Figure 11

Three- and two-dimensional difference Fourier maps of the Ge1 atom
parallel to the (001) plane at z/c = 0.2074 of τ3-Ba5Rh15Ge35.75.

[Figure ID: fig12]

Figure 12

Crystal structure of τ3-Ba5Rh15Ge35.75 and the coordination polyhedra
for the three different
Ba positions. Ba atoms are colored blue and Rh yellow, and Ge atoms
are colored pink, red [Ge1a (Ge1b/c not plotted)], and orange (Ge2).

[Figure ID: fig13]

Figure 13

DFT-derived enthalpy
of formation for Ba8MxGe46–x(M = Rh,
Pd, Ag; x = 0–6) and for the Ge46 framework. The information on the maximum solubility for Ba8PdxGe46–x and Ba8AgxGe46–x are taken from refs (40) and (29), respectively.

Electrical resistivity ρ and Seebeck coefficient S of κI-Ba8Rh1.2Ge42.8□2.0 as a function of the temperature.
The solid black line represents a linear fit to the S data.

[Figure ID: fig16]

Figure 16

Electrical resistivity ρ as a function of the temperature
for τ1-BaRhGe3. The solid line displays
the results of a least-squares fit to the Bloch–Grüneisen
model.

[Figure ID: fig17]

Figure 17

Electrical resistivity
ρ as a function of the temperature
and magnetic field (in Tesla) for τ2-Ba3Rh4Ge16. The solid line represents a least-squares
fit to the parallel resistor model. The inset shows low-temperature
details of the superconducting transition.

[Figure ID: fig18]

Figure 18

Temperature-dependent upper critical magnetic field Hc2 of τ2-Ba3Rh4Ge16. The solid line is a fit according to the
WHH model
(see the text).

[Figure ID: fig19]

Figure 19

Cp/T as a function
of the temperature and magnetic field for τ2-Ba3Rh4Ge16. The inset shows a detailed
view of the jump at the critical temperature (TC).

[Figure ID: fig20]

Figure 20

Electrical resistivity
ρ as a function of the temperature
for τ3-Ba5Rh15Ge35.75. The solid line represents a least-squares fit to the model described
in the text.